Answer
$y=x^2-6x+8$
Work Step by Step
We have to determine $a,b,c$ so that the graph of the function $y=ax^2+bx+c$ passes through the points $(1,3),(3,-1),(4,0)$.
Use the fact that each of the given points $(x,y)$ satisfies the equation $y=ax^2+bx+x$.
We find the system:
$\begin{cases}
a(1)^2+b(1)+c=3\\
a(3)^2+b(3)+c=-1\\
a(4)^2+b(4)+c=0
\end{cases}$
$\begin{cases}
a+b+c=3\\
9a+3b+c=-1\\
16a+4b+c=0
\end{cases}$
We will use the addition method. Multiply Equation 1 by -1 and add it to Equation 2 and Equation 3 to eliminate $c$:
$\begin{cases}
-a-b-c=-3\\
9a+3b+c=-1\\
16a+4b+c=0
\end{cases}$
$\begin{cases}
9a+3b+c-a-b-c=-1-3\\
16a+4b+c-a-b-c=0-3
\end{cases}$
$\begin{cases}
8a+2b=-4\\
15a+3b=-3
\end{cases}$
Simplify:
$\begin{cases}
4a+b=-2\\
5a+b=-1
\end{cases}$
Multiply Equation 1 by -1 and add it to Equation 2 to eliminate $b$ and determine $a$:
$\begin{cases}
-4a-b=2\\
5a+b=-1
\end{cases}$
$-4a-b+5a+b=2-1$
$a=1$
Determine $b$:
$4a+b=-2$
$4(1)+b=-2$
$4+b=-2$
$b=-6$
Determine $c$ by substituting $a,b$ in Equation 1:
$a+b+c=3$
$1-6+c=3$
$-5+c=3$
$c=8$
The system's solution is:
$(1,-6,8)$
The function is fully determined:
$y=x^2-6x+8$
